We next define two important subspaces of \(L^2(E,\varepsilon )\) . Given a weighted graph \((G,\upsilon ,\varepsilon )\) , the cycle space is \(\operatorname{Cyc}(E,\varepsilon ) = \ker \partial \)  and the cut space is \(\operatorname{Cut}(E,\varepsilon ) = \operatorname{im}\nabla \) . See [1]} for a treatment of these spaces in the unweighted case. Since \(\partial \)  and \(\nabla \)  are adjoint by Proposition REF , the cycle space and the cut space are orthogonal complements.
